A Model for Computational Science Investigations

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A Model for Computational Science Investigations

• AiS Challenge
• STI 2004
• Richard Allen

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Computational Science?

• Computational science seeks to gain an
  understanding of science through the use of
  mathematical models on supercomputers.

Computational Science involves teamwork
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Computational Science

• Complements, but does not replace, theory and
  experimentation in scientific research.

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Computational Science

• Is often used in place of experiments when
  experiments are too large, too expensive, too
  dangerous, or too time consuming.
• Can be useful in what if studies e.g. to
  investigate the use of pathogens (viruses,
  bacteria, fungi) to control an insect population.
  
• Is a modern tool for scientific investigation.

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Computational Science

• Has emerged as a powerful, indispensable tool
  for studying a variety of problems in scientific
  research, product and process development, and
  manufacturing.

• Seismology
• Climate modeling
• Economics
• Environment
• Material research

• Drug design
• Manufacturing
• Medicine
• Biology

Analyze - Predict
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Example Industry ?

• First jetliner to be digitally designed,
  "pre-assembled" on computer, eliminating need for
  costly, full-scale mockup.
• Computational modeling improved the quality of
  work and reduced changes, errors, and rework.

www.boeing.com/commercial/ 777family/index.html
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Example Roadmaps of the Human Brain

• Cortical regions activated as a subject remembers
  the letters x and r.
• Real-time MRI techno-logy may soon be
  incor-porated into dedicated hardware bundled
  with MRI scanners allowing the use of MRI in drug
  evaluation, psychiatry, neurosurgical planning.

www.itrd.gov/pubs/blue00/hecc.html
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Example Climate Modeling

• 3-D shaded relief representation of a portion of
  PA using color to show max daily temperatures.
• Displaying multiple data sets at once helps users
  quickly explore and analyze their data.

www.itrd.gov/pubs/blue00/hecc.html
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Computational Science Process
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Real World Problem

• Identify Real-World Problem
• Perform background research,
  focus focus on a workable problem.
• Conduct investigations (Labs),
  if if appropriate.
• Learn the use of a computational tool C, Java,
  StarLogo, Excel, Stella, and Mathematica.
• Understand current activity and predict future
  behavior.

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Working Model

• Simplify ? Working Model
  Identify and select factors to
  describe important aspects of
  Real World Problem deter-
  mine those factors that can be
  neglected.
• State simplifying assumptions.
• Determine governing principles, physical laws.
• Identify model variables and inter-relationships.

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Mathematical Model

• Represent ? Mathematical
  Model Express the Working
  Model in mathematical terms
  write down mathematical
  equations or an algorithm
  whose solution describes the
  Working Model.
• In general, the success of a mathematical model
  depends on how easy it is to use and how
  accurately it predicts.

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Computational Model

• Translate ? Computational
  Model Change Mathema-
  tical Model into a form suit-
  able for computational
  solu- tion.
• Computational models include languages, such
  as C or Java, or software, such as StarLogo,
  Stella, Excel, or Mathematica.

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Results/Conclusions

• Simulate ? Results/Con-
  clusions Run Computational
  Model to obtain Results draw
  Conclusions.
• Verify your computer program use check cases
  explore ranges of validity.
• Graphs, charts, and other visualization tools are
  useful in summarizing results and drawing
  conclusions.

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Real World Problem

• Interpret Conclusions
  Compare with Real World
  Problem behavior.
• If model results do not agree with physical
  reality or experimental data, reexamine the
  Working Model (relax assumptions) and repeat
  modeling steps.
• Often, the modeling process proceeds through
  several cycles until model isacceptable.

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Computational Science Process
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Computational Science Investigations

• A Computational science investigation should
  include
• An application - a scientific problem of interest
  and the components of that problem that we wish
  to study and/or include.
• Algorithm - the numerical/mathematical
  repre-sentation of that problem, including any
  numerical methods or recipes used to solve the
  algorithm.
• Architecture a computing platform and software
  tool(s) used to compute a solution set for the
  algorithm.

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Example A Falling Rock

• Determine the motion of a rock dropped from a
  height H, above the ground with initial velocity
  V.

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Working Model

• Governing principles d vt and v at.
• Simplifying assumptions
• Gravity is the only force acting on the body.
• Flat earth.
• No drag (air resistance).
• Model variables are H,V, g t, s, and v.

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Working Model (cont.)

• Form a discrete-in-time model to determine
  the position and velocity of the rock above
  the ground at equally spaced times, t0, t1,
  t2, , tn e.g. t0 0 sec, t1 1 sec t2
  2 sec, etc.
• v0 v1 v2
  vn
• s0 s1 s2
  sn
• _____________________________
• t0 t1 t2
  tn

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An Illustration
t time (in seconds)
0
1
2
3
4
100
90
80
70
60
s Displacement (in meters)
50
40
30
20
10
0
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An Illustration (cont.)
t time (in seconds)
0
1
2
3
4
100
90
80
70
60
s Displacement (in meters)
50
40
30
20
10
0
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An Illustration (cont.)
t time (in seconds)
0
1
2
3
4
100
90
80
70
60
s Displacement (in meters)
50
40
30
20
10
0
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An Illustration (cont.)
t time (in seconds)
0
1
2
3
4
100
90
80
70
60
s Displacement (in meters)
50
40
30
20
10
0
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An Illustration (cont.)
t time (in seconds)
0
1
2
3
4
100
90
80
70
60
s Displacement (in meters)
50
40
30
20
10
0
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Mathmatical Model

• Given an initial time, t0, an initial height,
  H, and an initial velocity, V, generate the time
  history of heights, sn, and velocities, v, by the
  formulas

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Computational Model

• Pseudo Code
• Input
• t0, initial time V, initial velocity H, initial
  height
• g, acceleration due to gravity ?t, time step
• imax, maximum number of steps
• Output
• ti, t-value at time step i
• si, height at time ti
• vi, velocity at time ti

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Example Falling Rock

• Initialize
• set ti t0 0 vi v0 V si s0 H
• print ti, si, vi
• Time stepping i 1, imax
• set ti ti ?t
• set si si vi?t
• set vi vi - g?t
• print ti, si, vi
• if (si lt 0), quit
  Excel Model

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Interpretation

• To create a more more realistic model of a
  falling rock, some of the simplifying assumptions
  could be dropped e.g., incor-porate drag -
  depends on shape of the rock, is proportional to
  velocity.
• Improve discrete model
• Approximate velocities in the midpoint of time
  intervals instead of the beginning.
• Reduce the size of ?t.

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A Virtual Science Laboratory

• The site below is a virtual library to visualize
  science. It has projects in mechanics,
  electricity and magnetism, life sciences, waves,
  astrophysics, and optics. It can be used to
  motivate the development of mathematical models
  for computational science projects.
• explorelearning

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Referenced URLs

• AiS Challenge Archive site        
• www.challenge.nm.org/Archive/
• Explorescience site
• www.explorelearning.com
• Boeing example
• www.boeing.com/commercial/777family/index.html
• Road maps for the human brain and climate
  modeling examples
• www.itrd.gov/pubs/blue00/hecc.html

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An interesting modeling site

• Formulating models
• www.cnr.colostate.edu/class_info/nr575/webfiles/L0
  5_Formulating_Continuous_Time_Models.pdf
• Falling bodies
• http//hypertextbook.com/physics/mechanics/falling
  /
• Master tools
• http//www.shodor.org/master/